Undoubtedly their earliest time-count was that common to primitive tribes everywhere—a measurement of the solar year by lunations or “moons.” The exact lunar month is 29 days, 12 hours, 44 minutes, 3 seconds; but primitive peoples usually estimate it at 28 days, and allow 13 months to the solar year, as do yet many North Asiatic peoples, and as probably did the early Aryans;[[25]] or, they estimate the “moon” at 30 days, and allow 12 moons to the year. There are good grounds for believing that the Mayan tribes were at one time divided in custom about this, some using one, some the other method. At the time of the Conquest they had undoubtedly reached a knowledge of the length of the year as 365 days; and there is considerable probability that some of them at least made the correction arranged for in our bissextile or leap year.[[26]]
This is all familiar enough and would create no difficulty in deciphering these aboriginal almanacs; but a disturbing element enters. The real time-count by which they adjusted the important events of their lives, and which is most prominent in their records, had nothing to do with the motions of the sun, or the moon, or any other natural phenomenon. It was based on purely mythical relations supposed to exist between man and nature. As the number 20 (fingers and toes) completes the man, and as all the directions, that is, potencies, of the visible and invisible worlds were held to be 13, these two numbers, 13 and 20, formed the basis of an astrological and ritual calendar, by which auspicious and inauspicious days were assigned, future events foretold, the major feasts and festivals of religious worship dictated, and the like.
This singular time-count of 20 × 13 = 260 days was adopted with slight variations by every semi-civilized nation of Mexico and Central America, and even the names of the 20 days are practically of the same meaning in all these languages.[[27]] It constituted the tonalamatl of the Nahuas, the “Book of Days,” used in divination.
This sacred period was subdivided into four equal parts of 65 days each, each of which was assigned to the rule of a special planet or star, and to a particular cardinal point with attendant divinities; and each was marked with a color of its own, white, black, red, or blue.
Each “month” of 20 days was subdivided into four periods of five days each, again each having its own divinity, assignment, etc.
But the importance to us of the tonalamatl is that its computations underlay the measurement of long periods of time, the less and greater cycles. These were estimated by the methods of the sacred year, in groups of 13, 20, 24, 52, 104, 260 years, etc. These irregular numbers had to be brought into unison with the lunar and solar years, with the vigesimal system of counting by 20 and its multiples, and with the observed motions of the planets, who were divinities controlling the ritual divisions of time.
To devise a mathematical method of equalities and differences by which these conflicting numbers could be placed in harmonious relations, subsumed under common measures, and the ceremonies and forecasts which they controlled assigned by uniform laws—this is the arithmetical problem which fills the pages of the Mayan Codices, and in parts or at length is spread over the surface of the inscribed monuments and painted vases. We need not search for the facts of history, the names of mighty kings, or the dates of conquests. We shall not find them. Chronometry we shall find, but not chronicles; astronomy with astrological aims; rituals, but no records. Pre-Columbian history will not be reconstructed from them. This will be a disappointment to many; but it is the conclusion toward which tend all the soundest investigations of recent years.
Let us recapitulate the numbers which the Maya mathematician had to deal with and adjust under some scheme of uniformity:—
| 1. | The “week” of 13 days, | 13. |
| 2. | The “month” of 20 days, | 20. |
| 3. | Its division into four parts (called tzuc), each, | 5. |
| 4. | The complete tonalamatl, 13 × 20 days, | 260. |
| 5. | Its divisions into four parts, each, | 65. |
| 6. | The solar year, counted as 18 months of 20 days each, | 360. |
| 7. | The solar year, counted as 12 months of 30 days each, | 360. |
| 8. | The solar year, counted as 13 lunar months of 28 days each, | 364. |
| 9. | The solar year, counted as 28 weeks of 13 days each, | 364. |
| 10. | The true solar year, days, | 365. |
| 11. | The bissextile year (?), | 366. |
| 12. | The apparent revolution of Venus (Noh-ek, the Great Star), days, | 584. |
| 13. | The apparent revolution of Mercury (?), days, | 115. |
| 14. | The apparent revolution of Mars (?), days, | 780. |
| 15. | The kin katun, or day-cycle of years, | 13. |
| 16. | The older cycle of years, | 20. |
| 17. | The newer cycle of years, | 24. |
| 18. | The katun cycle of years, | 52. |
| 19. | The double cycle of years, | 104. |
| 20. | The great cycle of years, | 260. |